#+TITLE: Ääriarvojakaumien luokittelun formalisointi
#+SUBTITLE: Affiinit reaaliakselin muunnokset
#+AUTHOR: Joel Kronqvist
#+LANGUAGE: fi
#+startup: beamer
#+LaTeX_CLASS: beamer
#+LaTeX_CLASS_OPTIONS: [bigger]
#+OPTIONS: H:2
#+LATEX_HEADER: \usepackage{fontspec}
#+LaTeX_HEADER: \setmonofont[Scale=0.8]{DejaVu Sans Mono}
#+LaTeX_HEADER: \usetheme{Copenhagen}
#+LaTeX_HEADER: \usepackage[finnish]{babel}
#+LaTeX_HEADER: \newcommand{\R}{\mathbb{R}}
#+LaTeX_HEADER: \renewcommand{\vec}[1]{\mathbf{#1}}

* Formalisointi

** Esimerkki

*** Text                                                              :BMCOL:
:PROPERTIES:
:BEAMER_col: .4
:END:

Sample text

- Sample item
- Second item
  
*** Lean4 code                                                        :BMCOL:
:PROPERTIES:
:BEAMER_col: .6
:END:

#+NAME: leanexample
#+begin_src lean4
    variable (p q : Prop)
    
    lemma (h : p ∧ q) : q ∧ p :=
      have hp : p := h.left
      have hq : q := h.right
      show q ∧ p from
        And.intro hq hp
#+end_src


* Ääriarvojakaumat

** Mitä ne ovat?

Matikkaselitystä


** Kertymäfunktiot


\begin{align*}
  \Phi_\alpha (x) &:= \begin{cases}
    0, & \text{kun}\ x \le 0, \\
    \exp( -x^{-\alpha}), &\text{muutoin}, \\
  \end{cases} \\
  \Psi_\alpha (x) &:= \begin{cases}
    \exp(-(-x)^\alpha), & \text{kun}\ x \le 0, \\
    1, &\text{muutoin}, \\
  \end{cases} \\
  \Lambda(x) &:= \exp(-e^{-x}), x \in \R
\end{align*}


** Kuvaajat

#+name: plot-frechet
#+begin_src python :results output :file ./plot.png :exports results
  import numpy as np
  import matplotlib.pyplot as plt

  fig, axes = plt.subplots(1, 3)
  fig.set_size_inches(10, 4)

  a1 = 3.0
  a2 = 1.2

  pl = lambda axi, x, f, lab: axes[axi].plot(x, np.vectorize(f)(x), label=lab)

  pl(
      0,
      np.linspace(1e-6, 3, 500),
      lambda x: a1*x**(-a1-1)*np.exp(-(x**(-a1))) if x > 0 else 0,
      "$ \\frac{\\mathrm{d}}{\\mathrm{d} x} \\Phi_\\alpha (x) $"
  )

  pl(
      1,
      np.linspace(-3, 1, 500),
      lambda x: a2 * ((-x) ** (a2 - 1)) * np.exp(-1 * ((-x) ** a2)) if x < 0 else 0,
      "$ \\frac{\\mathrm{d}}{\\mathrm{d} x} \\Psi_\\alpha (x) $"
  )

  pl(
      2,
      np.linspace(-5, 5, 500),
      lambda x: np.exp(-x) * np.exp(-np.exp(-x)),
      "$ \\frac{\\mathrm{d}}{\\mathrm{d} x} \\Lambda (x) $"
  )

  for ax in axes:
      ax.legend()

  plt.savefig('plot.png')
#+end_src

#+RESULTS: plot-frechet

#+ATTR_LATEX: :width 1.1\textwidth
#+CAPTION: Test caption
[[./plot.png]]


* Affiinit muunnokset

** Dia 1

Teksti 1


* TODO
** Nimi?